2 Answers
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The online OED has a separate entry for off-duty, which can function as an adverb, a noun and an adjective. It is made up of the preposition off and the noun duty. In your example, it is perhaps best seen as an an adjective postmodifying time.

Yes - off duty is more cohesive than off school, say. But 'it' would be written as two orthographic words if used in the predicative position. Off work is a less ambiguous-to-analyse example; the compound out-of-work exists, but not, I think, off-work.
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Edwin AshworthNov 17 '12 at 9:40

Would you consider the at work in time at work to be an adjective post-modifying time? I think the prepositions in these cases are just that: prepositions.
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RobustoNov 17 '12 at 11:30

@Robusto: No. I think off-duty is a special case. But I agree that it can be analysed in more than one way.
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Barrie EnglandNov 17 '12 at 11:33

It's sometimes almost meaningless trying to reason by analogy with equivalent-looking expressions that seem more readily analysable. In He went off (He went off right after the end of the play sounds totally acceptable), off is certainly not an adjective (as it undoubtedly is in The cheese is off). I think that the only near-acceptable thing to do is to put these whatsits into a separate class. Those doing so usually call the class particles, but put such disparate elements (such as the infinitive-marker to, the negativiser not) into this new class they're re-inventing the adverb.
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Edwin AshworthNov 17 '12 at 11:49

@Robusto: Can you give a reference that would allow multi-word (as opposed to hyphenated-compound) adjectives? I certainly think that some multi-word constructions are best treated principally as single lexemes (eg ship of the desert, spick and span). I've seen the term 'adjectival' used for 'any expression doing the job of an adjective' - ie used for the superset - eg in 'The Grammar Plan Book'; Weaver - but it has confusingly differing usages.
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Edwin AshworthNov 17 '12 at 12:05

You've come across a much-debated, and probably still-unresolved, part of English semanto-syntactic (meaning + grammar) analysis here, Listenever.

The authors of CGEL would doubtless say that the usages of off in

He needs more time off.

and

He needs more time off work / school / duty. ...

are so similar that they should not be put into different classes. They use the traditional term preposition for the second usage, where off fulfils the syntactic role of relating the occupation 'he' needs to be resting from to the first part of the sentence (He needs more time), as well as meaning 'away from'. They use the term 'intransitive preposition' (not their coinage) for the usage which assumes the hearer can easily fill in the unstated occupation mentally.

However, Cappelle, in this paper, argues in my opinion convincingly that this particular (no pun intended) lumping of word-classes is erroneous.

+1 for the Cappelle. It’s an exaggeration to say that any English word can serve as any part of speech; but in these cases I see no reason not to say that these are prepositions acting as ... What? They’re compounded with verbs, nouns, &c: postfixed, but semantically bound like Latin preposition prefixes, and in some degree “separable”, like German preposition prefixes. Would “proxifix” serve?
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StoneyBNov 17 '12 at 12:41

1

Proxifixes will be a dustbin-class within weeks.
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Edwin AshworthNov 17 '12 at 14:13

It would at least be a somewhat smaller and more manageable dustbin. Secondary component of a phrasal verb?
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StoneyBNov 17 '12 at 14:22

I've struck gold!English Club: Preposition Rule There is one very simple rule about prepositions. And, unlike most rules, this rule has no exceptions. Rule: A preposition is followed by a "noun". It is never followed by a verb. // So there! (This will keep me laughing over Christmas.)
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Edwin AshworthDec 19 '14 at 11:22